# Finding flux

1. May 1, 2009

### -EquinoX-

1. The problem statement, all variables and given/known data

Find the flux of the vector field through the surface of the closed cylinder of radius c and height c, centered on the z-axis with base on the xy-plane.

2. Relevant equations

3. The attempt at a solution

Can I just use the divergence theorem here? Find the divergence of the vector field and then multiply it by the volume of the cylinder?

2. May 1, 2009

### HallsofIvy

Staff Emeritus
Yes, you can use the divergence theorem but the is not "find the divergence of the vector field and then multiply it by the volume of the cylinder" unless the divergence is a constant. By the divergence theorem, the integral over the surface is the integral of the divergence over the cylinder.

3. May 1, 2009

### -EquinoX-

so if it's not a constant then I should integrate it right

4. May 1, 2009

### HallsofIvy

Staff Emeritus
Yes!

5. May 2, 2009

### -EquinoX-

if I have:

$$\vec{F} = 6x\vec{i} + 6y\vec{j}$$

then according to greens theorem this is 0 right? so therefore the flux is 0 as well?

6. May 3, 2009

bump!

7. May 3, 2009

### dx

No, it's not zero. You don't use green's theorem in this problem. The theorem you have to use is Gauss' divergence theorem. First find the divergence of F. What is the divergence of (6x, 6y)?

8. May 3, 2009

### -EquinoX-

it's just 12... and then multiply it by the volume right, which is $$\pi c^3$$ so it's $$12\pi c^3$$

9. May 3, 2009

### dx

That's right. Now just use the divergence theorem. The flux out of the cylinder is ∫(div F)dV over the volume of the cylinder.

10. May 3, 2009

### -EquinoX-

Last edited by a moderator: May 4, 2017
11. May 3, 2009

### dx

What limit?

12. May 3, 2009

### -EquinoX-

read my attachment above.. I don't understand the question it self

13. May 3, 2009

### dx

14. May 3, 2009

### -EquinoX-

so the limit and the flux density are both 12?

15. May 3, 2009

### dx

If by 'flux density' you mean 'divergence', then yes.

16. May 3, 2009

### -EquinoX-

yea flux density is divergence